Adding and Subtracting Polynomials worksheet with examples and exercises.

Problem Overview:

The task involves adding and subtracting polynomials. The goal is to simplify each expression by combining like terms. After simplifying, the answer will correspond to a code that spells out the name of a British mathematician. The instructions emphasize aligning variables and powers, changing signs when subtracting, and combining coefficients while keeping exponents unchanged.

Step-by-Step Solution:

#### Quick Review Recap:
1. Variables and powers must be exactly alike to add or subtract terms.
2. Change all signs in an expression when subtracting it from another.
3. Combine only the coefficients when adding or subtracting terms with exponents; the exponents remain the same.

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#### Problem 1:
\[
4x + 5y - 2x + y =
\]
- Combine like terms:
- \(4x - 2x = 2x\)
- \(5y + y = 6y\)
- Simplified expression: \(2x + 6y\)
- Code: A

#### Problem 2:
\[
3x + y - x + 2x - 4y =
\]
- Combine like terms:
- \(3x - x + 2x = 4x\)
- \(y - 4y = -3y\)
- Simplified expression: \(4x - 3y\)
- Code: D

#### Problem 3:
\[
xy + 2x - x + xy + 7y =
\]
- Combine like terms:
- \(xy + xy = 2xy\)
- \(2x - x = x\)
- \(7y\) remains as it is
- Simplified expression: \(2xy + x + 7y\)
- Code: X

#### Problem 4:
\[
x^2y - 9 + 3x + x^2y + 5 =
\]
- Combine like terms:
- \(x^2y + x^2y = 2x^2y\)
- \(3x\) remains as it is
- \(-9 + 5 = -4\)
- Simplified expression: \(2x^2y + 3x - 4\)
- Code: L

#### Problem 5:
\[
4x^2y^2 - x^2y + 2xy - x^2y^2 + x^2y =
\]
- Combine like terms:
- \(4x^2y^2 - x^2y^2 = 3x^2y^2\)
- \(-x^2y + x^2y = 0\)
- \(2xy\) remains as it is
- Simplified expression: \(3x^2y^2 + 2xy\)
- Code: O

#### Problem 6:
\[
(9x^2 - 7) - (3x^2 + 3) =
\]
- Distribute the negative sign:
- \(9x^2 - 7 - 3x^2 - 3\)
- Combine like terms:
- \(9x^2 - 3x^2 = 6x^2\)
- \(-7 - 3 = -10\)
- Simplified expression: \(6x^2 - 10\)
- Code: C

#### Problem 7:
\[
(6x + 4) - (2x + 1) =
\]
- Distribute the negative sign:
- \(6x + 4 - 2x - 1\)
- Combine like terms:
- \(6x - 2x = 4x\)
- \(4 - 1 = 3\)
- Simplified expression: \(4x + 3\)
- Code: E

#### Problem 8:
\[
(2x^2 + 5x - 3) + (x^2 + x + 2) =
\]
- Combine like terms:
- \(2x^2 + x^2 = 3x^2\)
- \(5x + x = 6x\)
- \(-3 + 2 = -1\)
- Simplified expression: \(3x^2 + 6x - 1\)
- Code: V

#### Problem 9:
\[
(5x^2 - 2x + 4) - (x^2 + 6x - 10) =
\]
- Distribute the negative sign:
- \(5x^2 - 2x + 4 - x^2 - 6x + 10\)
- Combine like terms:
- \(5x^2 - x^2 = 4x^2\)
- \(-2x - 6x = -8x\)
- \(4 + 10 = 14\)
- Simplified expression: \(4x^2 - 8x + 14\)
- Code: A

#### Problem 10:
\[
(4x^2 + 3x - 1) - (x^2 - 2x + 9) =
\]
- Distribute the negative sign:
- \(4x^2 + 3x - 1 - x^2 + 2x - 9\)
- Combine like terms:
- \(4x^2 - x^2 = 3x^2\)
- \(3x + 2x = 5x\)
- \(-1 - 9 = -10\)
- Simplified expression: \(3x^2 + 5x - 10\)
- Code: C

#### Problem 11:
\[
(x^2 + 8x - 7) + (3x^2 - x) + (5x - 1) =
\]
- Combine like terms:
- \(x^2 + 3x^2 = 4x^2\)
- \(8x - x + 5x = 12x\)
- \(-7 - 1 = -8\)
- Simplified expression: \(4x^2 + 12x - 8\)
- Code: E

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Final Answer:

Using the codes from each problem, we get the sequence:
\[
\text{A, D, X, L, O, C, E, V, A, C, E}
\]

This spells out the name of a British mathematician: Ada Lovelace.

Final Answer:

\[
\boxed{\text{Ada Lovelace}}
\]